magnified by a factor of n to recover the size of the original figure; again, let n = 2 as in our bisection case, so that the bisected square contains 22 = 4 copies of the original figure, each of which must be doubled in size to recover the area of the original figure. Log(4)/log(2) = 2, so the square is two-dimensional. So far so good. It’s worth pointing out that in these more familiar cases intuitive dimension = topological dimension = fractal dimension. That is not the case for all figures, though.
Finally, consider our broccoli-like fractal: the Pythagoras Tree. The Pythagoras Tree, as you can easily confirm, has a fractal dimension of 2: at each step n in the generation, there are 2n copies of the figure present: 1 on the zeroth iteration, 2 after a single iteration, 4 after two iterations, 8 after three, 16 after four, and so on. Additionally, each iteration produces figures that are smaller by a factor of √2/2. Following our formula from above, we can calculate log(2)/log(√2/2), which is equal to 2. This accords with our intuitive ascription of dimensionality (the Pythagoras Tree looks like a plane figure) but, more interestingly, it fails to accord with the topological dimension of the figure. Perhaps surprisingly, the Pythagoras Tree’s topological dimension is not 2 but 1—like a simple curve, it can be covered by disks such that no point is in the intersection of more than two disks. Topologically, the Pythagoras Tree behaves like a simple one-dimensional line, while in other ways it behaves more like a higher dimensional figure. Fractal dimension lets us quantify the amount by which these behaviors diverge: in fact, this is a characteristic that’s common to many (but not all) fractals. In addition to the two-pronged “fine detail and self-similarity” definition given above, Mandelbrot, in his
- The math behind this assertion is, again, beyond the scope of what we’re concerned with here. For a detailed discussion of why the topological dimension of fractal canopies—the class of figures to which the Pythagoras Tree belongs—is 1, see Mandelbrot (1986), Chapter 16.